Alain Badiou is a highly original, indeed decidedly iconoclastic thinker whose work has ranged widely over areas of equal concern to philosophers in the ‘continental’ and mainstream analytic traditions. These areas include ontology, epistemology, ethics, politics, and – above all – philosophy of mathematics. It is unfortunate, and symptomatic of prevailing attitudes, that his work has so far receivedminimal attention from commentators in the analytic line of descent. Here I try to help the process of reception along by describing Badiou’s remarkably ambitious approach to issues of mathematical (more specifically: of set-theoretical) ontology, and by explaining just where his project stands in relation to some major issues within current analytic debate. Chief among them are: the issue between realists and anti-realists – along with various avowed middle-ground or compromise solutions – and those oddly tenacious problems-from-Wittgenstein (e.g., concerning what it means to follow a rule) that have so preoccupied philosophers over the past decade. In particular I stress the unusual, indeed unique combination in his thought of high formal rigour and conceptual clarity allied to a speculative scope and inventiveness which tend to make those other discussions appear somewhat self-absorbed and parochial. Most importantly, Badiou engages these issues at a level of creative as well as of technical or analytic grasp, which puts his thinking closely in touch with the way that set theory has itself evolved through a constant process of – in Badiou’s phrase – ‘turning paradoxes into concepts.’ I also discuss his strong and principled rejection of the ‘linguistic turn’ in its manifold (analytic and continental) variants, and his idea of the ‘event’ as that which inherently eludes or surpasses the conceptual resources of any received ontology, whether in mathematics and the natural sciences or in the history of genuinely epochal changes in politics and ethics. All in all, I put the case for Badiou as a thinker of the first importance not only for the impressive range, depth and originality of his work, but also because it points to an escape-route from some of the more cramped or windowless quarters of present-day philosophic thought